VbvAstE-001
Book Boris V. Vasiliev Astrophysics
Book Boris V. Vasiliev
Astrophysics
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Using the definition of a chemical potential of ideal gas (of particles with spin=1/2)<br />
[12]<br />
[ ( )<br />
Ne 2π<br />
2 3/2 ]<br />
µ e = kT log<br />
(2.8)<br />
2V m ekT<br />
we obtain the full energy of the hot electron gas<br />
[<br />
E e ≈ 3 ( )<br />
2 kT Ne 1 + π3/2 aBe 2 3/2<br />
]<br />
n e , (2.9)<br />
4 kT<br />
where a B =<br />
2<br />
m ee 2<br />
is the Bohr radius.<br />
2.1.3 The correction for correlation of charged particles<br />
in a hot plasma<br />
At high temperatures, the plasma particles are uniformly distributed in space. At this<br />
limit, the energy of ion-electron interaction tends to zero. Some correlation in space<br />
distribution of particles arises as the positively charged particle groups around itself<br />
preferably particles with negative charges and vice versa. It is accepted to estimate<br />
the energy of this correlation by the method developed by Debye-Hükkel for strong<br />
electrolytes [12]. The energy of a charged particle inside plasma is equal to eϕ, where<br />
e is the charge of a particle, and ϕ is the electric potential induced by other particles<br />
on the considered particle.<br />
This potential inside plasma is determined by the Debye law [12]:<br />
where the Debye radius is<br />
r D =<br />
ϕ(r) = e r e− r<br />
r D (2.10)<br />
(<br />
4πe 2<br />
kT<br />
−1/2<br />
∑<br />
n aZa) 2 (2.11)<br />
For small values of ratio<br />
r<br />
r D<br />
, the potential can be expanded into a series<br />
ϕ(r) = e r − e + ... (2.12)<br />
r D<br />
The following terms are converted into zero at r → 0. The first term of this series is<br />
the potential of the considered particle. The second term<br />
√ ( 3/2<br />
E = −e 3 π ∑<br />
N aZa) 2 (2.13)<br />
kT V<br />
a<br />
is a potential induced by other particles of plasma on the charge under consideration.<br />
And so the correlation energy of plasma consisting of N e electrons and (N e/Z) nuclei<br />
with charge Z in volume V is [12]<br />
a<br />
E corr = −e 3 √<br />
πne<br />
kT Z3/2 N e (2.14)<br />
17